12 problems

Planar pinned dot-product conjecture. If $\dim A>1$, then there exists $a\in A$ such that

Let $n\ge 3$ and let $A\subseteq \mathbb R$ be a Borel set. Write $\dim_H A$ for the Hausdorff dimension of $A$. Shifted-product nonempty-interior conjecture. For every $n\ge 3$, t…

Let $p$ be prime, let $X\subset\mathbb F_p^2$, and let $E_s(X)$ denote the exceptional set of projection directions with projected dimension at most $s$. Finite-field exceptional-s…

Cholak–et al. sharpness conjecture. The lower bounds obtained for $T_{n,k}(a,s)$ by Cholak et al. are sharp for all $k$. The source does not reproduce the proposed lower-bound form…

Let $t\in[0,3]$ and let $s\in\left[0,\min\{t,1\}\right]$. For a Borel set $A\subseteq\mathbb{R}^3$ with $\dim A=t$, write $\rho_\theta(A)$ for its restricted projection at angle…

Let $E\subseteq\mathbb{R}^d$ be a self-similar or self-affine set that is not contained in a hyperplane. It is Newhouse projectively thick if, for every invertible linear transform…

Let $I\subset\mathbb{R}$ be a compact interval, and let $\gamma:I\to S^2$ be a $C^2$ curve satisfying the escaping great circle condition … for every $\theta\in I$. Let…

Let $u$ be a near-$d$-regular measure on $mathbb{R}^{2}$, let $0leq s<d$, and let $mathsigma$ be a Borel probability measure on $S^{1}$. Suppose that for $mathsigma$-almost every…

Let $F_\alpha$ be the operator defined in the paper, let $\mathcal{I}$ be the index set of the construction, let $p_{\mathbf{i}}$ be the corresponding probabilities, and let $M$ de…

Projected-sum dimension conjecture. For almost every $\theta\in(0,2\pi)$,

Let $B\subset\mathbb{R}^{3}$ be an analytic set, and let $(\rho_{\theta})_{\theta\in U}$ and $(\pi_{\theta})_{\theta\in U}$ be non-degenerate families of projections onto lines and…

Let $E\subset\mathbb{R}^2$ have Hausdorff dimension $\beta\leq 1$. For $\omega\in S^1$, let $p_\omega(E)$ be the orthogonal projection of $E$ onto the line through the origin in th…